Wednesday, August 10, 2011

Pythagoras 1, myself a half

kw: book reviews, nonfiction, mathematicians, mathematics, history

Almost any article of advice to authors states that every equation found in a "popular" book will reduce the book's potential audience by half. Let's see, there seem to be multiple equations on almost every page of a 260-page book, so the potential audience, out of seven billions of humans, must be no more than one. In fact, if you start with all the atoms in the universe and divide by two 260 times, the result is very close to one!

I sometimes call myself a working mathematician. That doesn't mean I have any degrees in the subject, but I do use a lot of math in my work. Not just adding and subtracting, but calculus, trigonometry, and linear analysis. But I have long known my limitations. I hesitate to mention it, but it was the math needed for senior Physics that induced me to change my major to Geology one semester before graduating, delaying graduation by more than two years. Nonlinear analysis and non-Euclidean geometries defeated me.

Then there are proofs. I can understand the occasional proof, and I can even take delight in a particularly facile one, as yesterday's post shows. I did learn to derive certain algebraic relationships. Luckily, once my career hit its stride, tons of math resources showed up on the Internet, and I've been able to look up the "how to" of a great many derivations that are difficult for me.

I still enjoy reading books in the 510-519 section of the Dewey Decimal system, and math history hits that right in the middle. The book cataloged 516.22/K is Hidden Harmonies: The Lives and Times of the Pythagorean Theorem, by Robert Kaplan and Ellen Kaplan. It discusses in enormous detail the Theorem (often called just PT in the book), and shows dozens of ways it has been derived, or proved, or used in other proofs.

In fact, proofs abound, and these were my downfall. I read through the first half of the book, found myself slogging, and eventually skimmed through the second half, digging out nuggets here and there. Luckily, I slowed down long enough to at least get the flavor of a derivation that has puzzled me. In standard books about Pythagorean Triples (explained next), I had seen these formulas (here goes 15/16 of my readership):

Given positive integers U and V which are mutually prime, one even and the other odd, find a, b, and c thus:

a = U²-V²
b = 2UV
c = U²+V²

Then a, b and c will all be integers and obey the relationship c²=a²+b². Such a set of three integers is a Pythagorean Triple, and the most familiar is 3, 4, 5, because 9+16=25. Try U and V as 2 and 1:

a = 4-1 = 3
b = 2x2x1 = 4
c = 4+1 = 5

Now if U and V are 12 and 9,

a = 144-81 = 63
b = 2x12x9 = 216
c = 144+81 = 225

OK, that's about as far as I ever got with Pythagorean Triples (PTs). I made them too easy to find, so I didn't dig into it further. Of course, lots of other people dug into it, some for their whole lives. One such, Waclaw Sierpinski, asked whether there is or is not an unending number of PTs for which a and b and c are each triangular numbers. A triangular number is one found by adding n consecutive numbers, starting at 1, such as 1+2+3+4+5 = 15. So far, this PT, 8778, 10296, and 13530 has been found. No other examples are known. Sierpinski is best known for his "gasket" consisting of a triangle in which the middle quarter (a triangle upside-down from the first) has been removed, then each remaining triangle has been similarly dissected, and so on infinitely. The result is a fractal with no area left at all. The result of 0.75ⁿ approaches zero as n approaches infinity.

OK, I am delaying the inevitable. This kind of playing around the edges is about the best I can do. I bow before a couple who are consummate, real mathematicians, the kind who can make more math, for the rest of us to either use if we are sufficiently competent, or stand aside while the gods of the math world do things with it that are beyond our grasp. I'll go back to my toy box now.